We want to characterize which sequences of whole numbers $(n_m)$ admit a weakly
mixing transformation $\tau$ such that $\tau$ is rigid along $(n_m)$, and which times $(n_m)$ admit a weakly mixing transformation $\tau$ such that $\tau$ is not recurrent along $(n_m)$.
These questions are in opposition, but also related. The necessary properties are a mix of sparsity and combinatorial, and/or algebraic, structure of the sequence. Examples and counterexamples, as well as some general results, will be described. This talk is based on joint current work with V. Bergelson (Ohio State), A. del Junco (Toronto), and M. Lemanczyk (Torun).